Mathematics

Session Hyperbola Session - 2

Session Objectives

Session Objectives Equation of chord joining two points on the hyperbola Equation of chord whose mid-point is given Equation of pair of tangents from an external point Equation of chord of contact Asymptotes of hyperbola Rectangular hyperbola Equation of rectangular hyperbola referred to its asymptotes as the axes of coordinates Director circle

Equation of the Chord Joining Two Points on the Hyperbola The equation of the chord joining two points on the hyperbola is which reduces to

Equation of Chord whose Mid-Point is Given Equation of chord of the hyperbola whose middle point is is given by

Equation of Pair of Tangents From and External Point Equation of pair of tangents from the point to the hyperbola is i.e. , where

Equation Chord of Contact Equation of chord of contact of point with respect to the hyperbola is , i.e. T = 0

Asymptotes of Hyperbola An asymptote is a straight line, which meets the conic in two points both of which are situated at an infinite distance, but which is itself not altogether (entirely) at infinity.

To Find the Equation of the Asymptotes of the Hyperbola Let the straight line y = mx + c ... (i) meet the given hyperbola in points, whose abscissae are given by the equation or ... (ii)

To Find the Equation of the Asymptotes of the Hyperbola If the straight line (i) be an asymptote, both roots of equation (ii) must be infinite. Hence, the coefficients of x2 and x in the equation (ii) must be zero. We have Hence, and c = 0

To Find the Equation of the Asymptotes of the Hyperbola Substituting the values of m and c in y = mx + c, we get , i.e. The combined equation of the asymptotes is

Points to Remember A hyperbola and its conjugate hyperbola have the same asymptotes. The equation of the pair of asymptotes differ the hyperbola and the conjugate hyperbola by the same constant only, i.e Hyperbola – Asymptotes = Asymptotes – Conjugate Hyperbola The asymptotes pass through the centre of hyperbola. The bisectors of the angles between the asymptotes of the hyperbola are the coordinate axes (or axes of the hyperbola).

Points to Remember As we know that combined equation of asymptotes is and equation of hyperbola is Equation of pair of asymptotes and equation of hyperbola differ by a constant only. (Important)

Rectangular Hyperbola or Equilateral Hyperbola A hyperbola whose asymptotes are at right angles to each other is called a rectangular hyperbola. The equations of asymptotes of the hyperbola are given by The angle between two asymptotes is given by

Rectangular Hyperbola or Equilateral Hyperbola If the asymptotes are at right angles, then Thus, the transverse and conjugate axes of a rectangular hyperbola are equal. Cor: Eccentricity of rectangular hyperbola is

Equation of the Rectangular Hyperbola Referred to its Asymptotes as the Axes of Coordinates Referred to the transverse and conjugate axes as the axes of coordinates, the equation of the rectangular hyperbola is The equation of asymptotes of the hyperbola (i) is Each of these two asymptotes is inclined at an angle of with the transverse axis.

Equation of the Rectangular Hyperbola Referred to its Asymptotes as the Axes of Coordinates Now rotating the axes through an angle in clockwise direction, keeping the origin fixed, then the axes coincide with the asymptotes of the hyperbola and Putting the values of x and y in (i), we get

Equation of the Rectangular Hyperbola Referred to its Asymptotes as the Axes of Coordinates This is the transformed equation of rectangular hyperbola (i). Thus, equation of rectangular hyperbola when its asymptotes taken as coordinate axes is Cor: If equation of a rectangular hyperbola be then equation of its conjugate hyperbola will be

Parametric Form of Rectangular Hyperbola xy = c2 If ‘t’ is non-zero variable, the coordinates of any point on the rectangular hyperbola xy = c2 can be written The point is also called point ‘t’.

Equation of Chord Joining Points ‘t1’ and ‘t2’ The equation of the chord joining two points and of hyperbola xy = c2 is This is the required equation of chord.

Equation of Tangent in Different Forms Equation of tangent in point form of the hyperbola xy = c2 Equation of tangent in parametric form

Equation Normal in Different Forms Equation of normal in point form

Equation Normal in Parametric Forms Equation of normal in parametric form Note: The equation of normal at is a fourth degree equation in t. Therefore, in general four normal can be drawn from a point to the hyperbola xy = c2.

Point of Intersection of Tangents at ‘t1’ and ‘t2’ to the Hyperbola xy = c2 The equations to the tangents at the points ‘t1’ and ‘t2’ are By solving these equations, we get point of intersection of tangents. Coordinates of point of intersection of tangents at ‘t1’ and ‘t2’ is .

Director Circle The locus of intersection of tangents which are at right angles is called director circle of Hyperbola. To find the locus of the point of intersection of tangents which meet at right angles.

Director Circle Let (h, k) be their point of intersection. We have [By putting the value of (h, k) in equations (iii) and (iv)] If between (iv) and (v), we eliminate m, we shall have a relation between h and k, i.e. locus of (h, k). Squaring and adding these equations, we get

Director Circle Locus of (h, k) is This is the equation of director circle, whose centre is origin and radius is

Class Test

Class Exercise - 1 If the chord through the points on the hyperbola passes through a focus, prove that

Solution The equation of the chord joining is If it passes through the focus (ae, 0), then By componendo and dividendo, [Proved]

Class Exercise - 2 Chords of the hyperbola touch the parabola Prove that the locus of their middle points is the curve

Solution If (h, k) be the mid-point of the chord, then the equation of the chord is T= S1, If it touches the parabola y2 = 4ax, then [Condition for tangency for any line y = mx + c to the parabola]

Solution contd.. Locus of (h, k) is

Class Exercise - 3 Find the point of intersection of tangents drawn to the hyperbola at the points where it is intersected by the line lx + my + n = 0.

Solution Let (h, k) be the required point. Equation of chord of contact drawn from (h, k) to the hyperbola is T = 0 The given line is lx + my + n = 0 ... (ii) Equations (i) and (ii) represent same line

Solution contd.. Coordinates of the required point

Class Exercise - 4 Prove that the product of the perpendiculars from any point on the hyperbola to its asymptotes is equal to

Solution Let be any point on the hyperbola The equation of the asymptotes of the given hyperbola are Length of perpendicular from

Solution contd… Length of perpendicular from

Class Exercise - 5 The asymptotes of a hyperbola are parallel to lines 2x + 3y = 0 and 3x + 2y = 0. The hyperbola has its centre at (1, 2) and it passes through (5, 3). Find its equation.

Solution Asymptotes are parallel to lines 2x + 3y = 0 and 3x + 2y = 0 Equations of asymptotes are 2x + 3y + k1 = 0 and 3x + 2y + k2 = 0 As we know that asymptotes passes through the centre of the hyperbola. Here centre of hyperbola is (1, 2).

Solution contd.. The equations of asymptotes are 2x + 3y – 8 = 0 and 3x + 2y – 7 = 0 Equation of hyperbola is (2x + 3y – 8) (3x + 2y – 7) + c = 0 It passes through (5, 3). Equation of hyperbola is (2x + 3y – 8)(3x + 2y – 7) – 154 = 0 i.e. 6x2 + 13xy + 6y2 – 38x – 37y – 98 = 0

Class Exercise - 6 The chord PP´ of a rectangular hyperbola meets asymptotes in Q and Q´. Show QP = P´Q´.

Solution Let equation of rectangular hyperbola is xy = c2. Equation of chord PP´ is It meets asymptotes, i.e. axes at Q and Q´respectively.

Solution contd.. [Proved]

Class Exercise - 7 The normal at the three points P, Q, R on a rectangular hyperbola, intersect at a point S on the curve. Prove that centre of the hyperbola is the centroid of PQR.

Solution Let equation of the rectangular hyperbola is xy = c2. Let ‘t’ be the parameter of any of points P, Q, R so that normal is ... (i) It passes through a point S on the hyperbola. Let coordinates of point

Solution contd.. (Remember this result) This is a cubic equation in t, and gives us the parameters of the three points P, Q, R, say If (h, k) is the centroid of ,

Solution contd.. Hence, centroid is (0, 0) which is centre of the hyperbola.

Class Exercise - 8 A rectangular hyperbola whose centre is C is cut by any circle of radius r in four points P, Q, R and S. Prove that

Solution Let the equation of rectangular hyperbola is xy = k2 ... (i) and equation of circle is where ... (iii) Eliminating y between equations (i) and (ii), we get

Solution contd.. This is biquadratic equation in x, which gives us the abscissae of the four points of intersection. Let x1, x2, x3, x4 are the roots of the equation. Similarly, eliminating x from (i) and (ii), we get

Solution contd..

Class Exercise - 9 Prove that the locus of the mid-points of the chords of the hyperbola which pass through a fixed point is a hyperbola whose centre is

Solution Let (h, k) be the coordinates of mid-point of the chord. Equation of chord is T = S1 It passes through a fixed point .

Solution contd.. Locus of (h, k) is This is an equation of hyperbola whose centre is

Class Exercise - 10 From a point, tangents to the rectangular hyperbola are drawn and they intersect each other at an angle of 45o. Prove that the locus of the point is the curve

Solution Here equation of rectangular hyperbola is ... (i) Equation of any tangent to this hyperbola is If it is passes through (h, k), then Let m1, m2 be the roots of the above equation, which gives the slopes of two tangents passing through (h, k).

Solution contd.. Given that angle between two tangents are 45o. Locus of (h, k) is

Thank you